In this paper, we construct a linkable ring signature scheme with unconditional anonymity. It has been regarded as an open problem in [22] since 2004 for the construction of an unconditional anonymous linkable ring signature scheme. We are the first to solve this open problem by giving a concrete instantiation, which is proven secure in the random oracle model. Our construction is even more efficient than other schemes that can only provide computational anonymity. Simultaneously, our scheme can act as an counterexample to show that [19, Theorem 1] is not always true, which stated that linkable ring signature scheme cannot provide strong anonymity. Yet we prove that our scheme can achieve strong anonymity (under one of the interpretations).